Isaac & O’Connor’s Discontinuity Theory

This is outside my comfort zone in many respects, but I was asked to comment on its relationship to the topic of this blog.

Glancing at it, it looks very logical, seems to have some helpful illustrations, to be more accessible than my usual sources, and to be backed up by scientific experiments (rather than just experience in practice). They key difference is that I am used to considering ‘difficult problems’ where the discussion is inevitably difficult, whereas this is looking at child development, so that we adults can understand the child’s difficulties and hence come to appreciate our own by extension. So the paper promises much. It seeks to build on Piaget, which seems a reasonable start.

Unfortunately, I gather that this body of work had a mixed reception and failed to have any institutionalised impact, only surviving in the memory of some students who haven’t been able to communicate the lessons where they are needed. (This will become clearer later.)

Keynes rightly made a distinction between mathematics proper and pseudo mathematics, similar to the distinction that some make between science proper and scientism. Unfortunately, as soon as I try to interpret the paper logically the content seems very pseudy. The textural description, graphics and formulae all seem very credible and pertinent, but I failed to reconcile them. But the role of a mathematician is not just to point out error, but to try to fix it, and my experience is that meaningless junk can sometime obscure genuine insight, just as meaningful (but misleading) scientism can obscure a lack of insight or even a dangerous ideology.

The most obvious comparator to make is with Whitehead’s process logic, and this is what I was asked about. But upon reflection I think it helpful to start more simply.

In classical type theory one has type 0 predicates (e.g. ‘objects’) that have no relata (things being predicated). For example, ‘the cat’. One then defines higher types of predicates recursively. Thus type n+1 predicates have type n relata. (Strictly, the relata are of type n or less, and at least one is of type n.)

A variation on the approach is to suppose that type 0 predicates relate themselves. Thus we either say that cats ‘just are’, or that cats just look after themselves. Dogs on the other hand seem to be born with a slot for a ‘pack leader’ relata, so they are of type 1.

Thus we have, for example:

0: The cat is being a cat.

1: The cat is hunting mice

2: The mother cat is teaching the kitten to chase mice.

and so on. This corresponds to my interpretation of the paper’s ‘system of levels’:

[A] series of increasingly complex structures is built up to serve as the basis of a stage/level system for psychological development. Each of these structures expresses a discrete extension of the proceeding structure. This discrete extension occurs through the introduction of one or more relations. With this discrete extension of structure a transformation of the self and the objects of the preceding structure takes place.

Actually, the paper is easiest to interpret from the kitten’s perspective:

2′: The kitten is chasing mice guided by the mother.

An alternative would be:

2”: The mother and kitten are chasing mice, with the mother as the teacher.

One needs to adapt the notation somewhat to have the kitten chase the mouse without direct involvement of the mother, and it does not seem to be the intention to cover this case. Thus the paper’s theory seems to be a strict specialization of type theory. It is not clear if this restriction is deliberate or not.

The paper very briefly reports some results that indicate that most 7 year olds can operate at level 2, and some 15 year olds can go beyond, to level 3, as in:

3: The kitten is chasing mice while signalling to its mother, with the manner of chasing and the signalling being related. (For example, while the mother is trying to guide the kitten, the kitten is trying to guide the mother into believing that it can chased mice adequately.)

From 17 some people apparently go a stage further:

4: The kitten is chasing mice while signalling to its two parents (as above), with the relationships with the two parents being inter-dependent.

and sometimes even a level beyond this, somewhat like:

5: The kitten is chasing mice while signalling to its two parents as above, while also giving off other signs (not directly concerned with chasing mice or the parents) to its two brothers, with the signs being different but inter-related. (Thus the relationship to the brothers structurally mirrors that to the parents.) At the same time, the way it is dealing with the parent ‘objects’ is related to the way that it is dealing with the brother ‘objects’. (Perhaps it does not want the parents to realise something about its relationship with its brothers, or vice-versa. Or perhaps it is just developing generic ‘social skills’).

This structure seems to me unnecessarily specific. One can easily envisage situations that are more complex than level 4 but do not meet the criteria for level 5. For example, where the kitten is chasing mice as in 4 while showing-off to its brothers without the brother relationship necessarily being at level 4. From a mathematical/logical point of view, the conceptual difficulty arises from the way the levels are constructed.

If we agree to call objects ‘level 0’ then in the paper’s scheme:

level 1 can relate to level 0

level 2 can also relate level 1 to level 0,

level 3 can also relate level 1 to level 1.

That which is added at each level may be called ‘the new mode’. The new mode of level 4 is then the ability to relate what is new at level 3. It is not clear what is intended here. Consider the ability to relate level 3 to level 2. This does not imply a full level 4 capability. Perhaps it is a hypothetical intermediate level of development that never occurs in practice in children? Or perhaps the lower levels can be considered to be degenerate cases of the higher levels? In any case, the way that level 4 is derived from level 3 seems inconsistent with the way that level 2 is derived from level 1. Type theory would be more logical but according to the experimental results not discriminating enough for the author’s purpose. My suggestion is as follows.

We start with a predicate of type 0. A predicate with relata of type zero zero will be said to be of type (0, …. 0), with as many 0s as there are relata. Inductively, a predicate whose relata are of types a, b, … z will be said to be of type (a, b, … z). Thus a type is a string of brackets and 0s. For convenience we can use n to represent the type that is added by the paper’s level n. Then the paper’s levels progressively add:

(0)

(1, 0)

(1, 1)

(3,3)

(4,4)

Actually, the paper’s levels have the extra restriction that some of the elements are identified, but hopefully the above serves to to illustrates its idiosyncrasies. What about (3,2) or (0,0)?

My point is not to propose a concrete alternative to the theory of the paper, but simply to suggest that it might be a good idea to try to fix the theory so that it could be interpreted logically, and to give an example of what I mean by that. Having taken a mathematical approach to refine the theory there remains choices which depend on an understanding of the subject matter – in this case child development.

According to the abstract:

Problem-solving experiments and experiments involving loss of skill under increasing stress have been designed to test the theory, the results of which support the theory.

According to the conclusion:

The general scheme for psychological development … is as follows: For [the young] one component is present. [Later] a second component emerges, with a corresponding transformation of the other component, after which both components develop continuously. [Then] a third component emerges … .

… The theory having been substantiated … .

The details are not provided, but from a logical perspective theories are only ever substantiated to the extent that they have not been falsified, and my own experience is that there are often equally credible alternatives to accepted theories that simply have never been considered. From a logical point of view there are two obvious ways of varying the theory:

To suppose that the general scheme is correct, but that – as above – the details may be different.

To suppose that there are discontinuities for the reasons given, but that there may also be other discontinuities. For example, I may progress form (0) to (1,0) to (1,1,0), with both steps being discontinuous.

It might also be interesting to consider the degeneration of capability in the elderly. Does it follow a similar pattern?

Comments

The paper seems to make some important points, but they are confused by an inappropriate mathematisation, and perhaps the authors have confused themselves.